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Full text of "Scientific Papers - Vi"

that here also any value of - kU is admissible, and for the same reason as before, viz., that when n +kU= 0, dv/dy may be discontinuous. Suppose, for example, that there is but one place where n+kU=Q. We may start from either wall with v = 0 and with an arbitrary value of dv/dy and gradually build up the solutions inwards so as to satisfy (3)*. The process is to be continued on both sides until we come to the place where n + kU=0. The two values there found for v and for dvjdy will presumably disagree. But by suitable choice of the relative initial values of dvjdy, v may be made continuous, and (as has been said) a discontinuity in dv/dy does not interfere with the satisfaction of (3). If there are other places where U has the same value, dv/dy may there be either continuous or discontinuous. Even when there is but one place where n + kU = Q with the proposed value of n, it may happen that dv/dy is there continuous.
The argument above employed is not interfered with even though U is such that dU/dy is here and there discontinuous, so as to make d^U/dy* infinite. At any such place the necessary condition is obtained by integrating (3) across the discontinuity. As was shown in my former paper (loc. cit.\ it is
. fdv\      . idU\        A                       /Pv
A  -7-    A   ^r-   . V = 0,   ..................(6)
\dyj         \dyj
A being the symbol of finite differences; and by (6) the corresponding sudden change in dv/dy is determined.
It appears then that any value of  kU is a possible value of n. Are other real values admissible ? If so, n + k U is of one sign throughout. It is easy to see that if d* U/dy* has throughout the same sign as n + k U, no solution is possible. I propose to prove that no solution is possible in any case if n + kU, being real, is of one sign throughout.
If V be written for U + n/k, our equation (3) takes the form
Z7'-=-----v, = k*U'v,...........................(V)
dy*       dy*            '
or on integration with respect to y, Tr,dv      dU'
rr/    7                               /0\
.           u vdy, .................. (8)
dy       dy               Jo
where K is an arbitrary constant.   Assume v = U'v'; then
dv'     K      k*
reached.
Graphically, the equation directs us with what curvature to proceed at any point alreadyughout. It is easy to recognizeal velocities, the motion